Stronger word than "imitate"

emulate or impersonate.

emulate: to copy something achieved by someone else and try to do it as well as they have.

impersonate: to intentionally copy another person's characteristics, such as their behavior, speech, appearance, or expressions, especially to make people laugh.

source - Cambridge Dictionary

hope this helps you.

Especially in the context of theater, we can say methodize. As Wordhippo defines it, methodize is:

To perform a theatrical role in accordance with the principles of method acting.

Where method acting is defined as:

An acting technique in which the actor fully immerses themselves into the character they are playing.

or, in Merriam Webster:

capitalized: a dramatic technique by which an actor seeks to gain complete identification with the inner personality of the character being portrayed

As you described, it involves the incorporation of the person's personality completely. Skilled actors can methodize (almost) perfectly.

The word clone originally had a very specific scientific and technical meaning but now has an informal meaning defined by the Cambridge online dictionary as:

someone or something that looks very much like someone or something else:

with the example :

Most people saw her as just another blond-haired, red-lipped Marilyn Monroe clone.

You could, therefore describe someone who was copying someone else's dress style, manner of speech, hairstyle, physical presentation and so on as "a clone" of the other.

You do sometimes see this in corporate life where a junior with ambition models themself on a senior member of the organisation, possibly the owner of the company, thinking that by emulating the other person they can become as successful as they have been. However it is almost always unsuccessful and is usually a cause of amusement to the rest of the staff who will often refer to the junior as "the clone" of the senior person.

I’m not sure that “appropriate” is generally used in the context you’re looking for but that is the first thing that came to mind.

“He’s so unoriginal, it’s like he’s trying to replicate his father.”

Also duplicate, carbon-copy, re-create, reincarnate, etc.

I can't fully support articulating a potential for one person to capture and then express the persona of another, but how about:

Google Dictionary: Encapsulate - verb: encapsulate; 3rd person present: encapsulates; past tense: encapsulated; past participle: encapsulated; gerund or present participle: encapsulating

To: express the essential features of (something) succinctly; "the conclusion is encapsulated in one sentence"

It would have to be ...encapsulate and express his father's persona.